Adaptive chi-square tests

We consider minimax hypothesis testing problem $H_0$: $f=f_0$, $f_0(x)\equiv 1$ on a distribution density $f$ of i.i.d. observations $X_1,\dots,X_n$, $X_i\in[0,1]$, $n\to\infty$ versus alternative corresponding to smooth densities $f$ which are distant enough from $f_0$. A distance between $f_0$ and $f$ is measured in $L_p$-norm and a smoothness $\sigma$ of $f$ is measured in $L_q$-norm. A priory the values $\sigma,p,q$ are not fixed but satisfy to constraints $1\le p\le 2$, $p\le q$, $\sigma>0$. We show that optimal minimax rate is provided by test procedures which are based on the union of chi-square tests with increasing number of cells.